Deigmatolhy—a - Τμήμα Επιστήμης Υπολογιστών...
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9
9.1
, . , , , , , , , .
, ( ), . . , (.. pixel, ).
50, , . , - 0 1, . , . , / , , ! ! :-) ,, ( ) .
. ( 9.1), . , - (). , , - . , , , . 1. , , - . . -, . , . .
9.1: .
1 .
328
. Ts ( ) . . , fs (fs = 1Ts ), - . , CD , 44100Hz. , 44100 , 144100 ! , ! . ...
: ; ; ; ; ; , bits; Nyquist Shannon, 1928 , 1949 , . Shannon-Nyquist.
(- ) B Hz, ,
fs > 2B, (9.1)
.
Ts 2B(
Ts 2fmax x(nTs) (. ) sinc(), nTs, n Z!!! 9.8 sinc() ( ) ( ) ().
t
... ...
9.8: sinc.
, , - 9.9(). ( ) fs ( ). B fs B. , 9.9(), , . , , , 9.9(). , 9.9(), .
B < fs B fs > 2B (9.18)
Shannon. (- :-) ). . 9.9(), Shannon , . fs B . ( ) . , . fs ...
aliasing 4., 9.10.
Arect( tT
) ATsinc(fT) = ATsinc(tT) Arect
(
f
T
)= Arect
( fT
)(9.15)
4 ...
9. 333
-B B0 f-fs-fs-B -fs+B fs fs+Bfs-B
X(f)
-B B0 f
X(0)
X(0)/Ts
Xs(f)
0 f-fs fs
Xs(f)
-B B fs+Bfs-B-fs-B -fs+B
()
()
()
X(0)/Ts
9.9: :
9.2.3
1. , x(t) , . , X(f), ([B,B]). , , - , ! Haa(f), H(f) , anti-aliasing filter, , . .
2. H(f) ( 9.7). , , , . , . .
3. x(t) . ( ) , . Fourier
(t) 1
rect( t
)(9.19)
0, . ,
334
t0 x
s(t)
= x
[nT
s]
T s
X(0
)/T s
Xs(
f)
-BB
0f
-fs
f sf s
+B
f s-B
-fs-
B-f
s+B
x(t
)
0t
X(0
)
X(f
)
-BB
0f
X(f
)
-BB
0f
-fs
f sf s
+B
f s-B
-fs-
B-f
s+B
X(0
)
X(0
)/T s
Xs(
f)
-BB
0f
-fs
f sf s
+Bf s
-B-f
s-B
-fs+
B
H(f
) = T
srec
t(f/
f s)
T s
t0 x
s(t)
= x
[nT
s]
x(t
)
0t
.....
.
x (t
)
-fs/
2f s
/2
9.10
:
.
9. 335
.
4. Shannon . : x(t) X(f), X(f) [B,B]. y(t) = x3(t), Y (f) 6= 0 [3B, 3B]. , fs > 6B. fs = 2B. ; y(nTs). 9.11. ;
h(t) ____3
y(nTs) = x3(nTs)
9.11: .
5. [B,B] [fs/2, fs/2]; , - - fs/2, . , . :-)
6. Shannon; . - - . , , 20 kHz , - 10 kHz. 2 10 = 20kHz .
, 2025 kHz, fs > 2 {20 25} = 40 50kHz. 20 Hz 20 kHz. 2 20 = 40 kHz . , CD ( Sony 70) 44.1 kHz, 20 kHz , .
9.2.4
.
1:
x(t) = 3 cos(400t) + 5 sin(1200t) + 6 cos(4400t) (9.20)
fs = 4000 Hz x[n]. - , x[n].
: x(t) , fmax = 2200 Hz.
2fmax = 2 2200 = 4400 > fs (9.21)
336
Shannon , . , . :-)
t = nTs = n 14000 sec. t nTs. :
x[n] = 3 cos(400nTs) + 5 sin(1200nTs) + 6 cos(4400nTs) (9.22)
= 3 cos(n
10
)+ 5 sin
(12n40
)+ 6 cos
(44n40
)(9.23)
2:
h(t) =sin(2fct)
fst(9.24)
fM < fc < fs fM fs = 1Ts h[nTs] = [n] n, fc =fs2 .
( Dirac ) :
[n] =
1, n = 0,
0,
(9.25)
:,
h(t) =sin(2fct)
fst=
sin(fst)
(fst)= sinc(fst) (9.26)
fs = 1Ts , t nTs,
h[nTs] = sinc(
fsnTs
)= sinc
( 1Ts
nTs
) h[n] = sinc(n) (9.27)
, sinc(n) =sin(n)
n ,
sin(n) = 0 sin(n) = sin(k) (9.28)
n = k n = k, k Z (9.29)
n, , ,
h[n] =
sinc(0) = 1, n = 0,
sinc(n) = 0, n 6= 0
,
h[n] = [n] (9.30)
3:
x(t) . Fourier X(f), [B,B]. Nyquist :
() x(t)
9. 337
() x(t t0)
() x(t)ej2f0t
() x(t t0) + x(t+ t0)
()dx(t)
dt
() x(t)x(t)
() x(t) x(t)
: ( Nyquist) -. :
() B, Nyquist fs = 2B.
() x(t t0) X(f)ej2ft0 (9.31)
B, Nyquist fs = 2B.
() x(t)ej2f0t X(f f0) (9.32)
f0, f0+B, fs = 2(f0+B).
() x(t t0) + x(t+ t0) X(f)ej2ft0 +X(f)ej2ft0 = 2X(f) cos(2ft0) (9.33)
B, fs = 2B.
() dx(t)
dt j2fX(f) (9.34)
, B, fs = 2B.
() x(t)x(t) X(f) X(f) (9.35)
, , . [B B,B +B] = [2B, 2B]. Nyquist fs = 4B.
() x(t) x(t) X(f)X(f) = X2(f) (9.36)
Nyquist fs = 2B.
4:
Nyquist
() sinc2(100t)
() 1100 sinc2(100t)
() sinc(100t) + 3sinc2(60t)
() sinc(50t) sinc(100t)
: Nyquist Shannon, .
338
2fmax ., . Fourier,
x(t) X(f) = X(t) x(f) (9.37)
sinc() , ( A = 1),
rect( t
T
) Tsinc(fT) (9.38)
T sinc(tT) rect(f
T
)= rect
( fT
)(9.39)
tri( t
T
) Tsinc2(fT) (9.40)
T sinc2(tT) tri(f
T
)= tri
( fT
)(9.41)
f f sinc(), rect() .
()
sinc2(100t) 1100
tri( f
100
)(9.42)
tri() [100, 100] Hz. 100 Hz, Nyquist fs = 200 Hz.
() , fs = 200 Hz.
()
sinc(100t) + 3sinc2(60t) 1100
rect( f
100
)+ 3
1
60tri( f
60
)(9.43)
60 Hz, Nyquist fs = 120 Hz.
()
sinc(50t) sinc(100t) 150
rect( f
50
) 1100
rect( f
100
)(9.44)
25 Hz, Nyquist fs = 50 Hz.
